All Questions
Tagged with ds.dynamical-systems sequences-and-series
32 questions
4
votes
2
answers
165
views
Convergence of the Cesàro mean of iterated continuous functions
Does anyone have a counter-example of the following statement :
Let $f : [0;1] \to [0;1]$ a continuous function w.r.t. the usual topology. Let $A_n(x) = \frac{1}{n} \sum_{k=0}^{n-1} f^k(x)$ for $n \ge ...
- 43
1
vote
0
answers
84
views
Coarse well-distributedness/equidistribution of Pell sequence prefixes
I am interested in the distributedness or "mixing" behavior of certain
linear recurrences modulo powers of $2$.
In particular, consider the Pell sequence (https://oeis.org/A000129),
modulo $...
- 11
13
votes
2
answers
555
views
Convergence of the sequence $s_{n+1}=s_n^2-s_{n-1}^2$
$s_{n+1}=s_n^2-s_{n-1}^2$, $s_0=\sqrt{x}$, $s_1=x$
This sequence seems simple, but is pretty confusing. If you try it with integers, you might think that it always diverges to infinity, but if you try ...
- 467
8
votes
2
answers
2k
views
5n+1 sequence starting at 7
Consider the following variant of the Collatz function: $f:\mathbb N\rightarrow\mathbb N$ is defined by
\begin{equation}
f(n):=\begin{cases}
n/2 & \text{if $n$ is even}\\
5n+1 & \...
- 654
8
votes
2
answers
340
views
Does $x_0=1/3$ lead to periodicity in the logistic map $x_{k+1}=4x_k(1-x_k)$?
Does $x_0=1/3$ lead to periodicity in the logistic map $x_{k+1}=4x_k(1-x_k)$?
I believe it does not, but this is equivalent to proving that $(2\pi)^{-1}\arcsin(\sqrt{1/3})$ is irrational. I am ...
- 3,098
7
votes
0
answers
429
views
Dynamics of a curious bijection of $\mathbb N$
The two sequences A48680 and A48679 of the OEIS define two mutually inverse bijections on the set of all strictly positive natural numbers given (for the comfort of the reader) as follows:
Given an ...
- 17.9k
4
votes
0
answers
114
views
Is this pair of coupled sequences known, and what are their properties?
I was examining the following pair of 'coupled' sequences (I don't know the correct terminology):
$a_{n+1}=a_n+b_n+\frac{a_n}{b_n}$
$b_{n+1}=b_n\left(1+\frac{b_n}{a_n}\right)$
Both sequences grow ...
12
votes
1
answer
992
views
Strange behavior of $x_{n+1}=x_n +\lambda \sin x_n$
Consider a sequence $(x_n)$ satisfying $x_{n+1}=x_n +\lambda \sin x_n$.
You would expect the sequence $x_n$ to depend on $x_0$ and to exhibit a chaotic, Brownian-type behavior, and indeed it does ...
- 3,098
25
votes
2
answers
2k
views
Do these rational sequences always reach an integer?
This post comes from the suggestion of Joel Moreira in a comment on An alternative to continued fraction and applications (itself inspired by the Numberphile video 2.920050977316 and Fridman, ...
1
vote
0
answers
189
views
Vandermonde shift
I'm looking for any known results on a shift operator commutated by a Vandermonde matrix. That is, let
$$T=\begin{bmatrix}0 & 1 & 0 & 0 & \cdots \\
0 & 0 & 1 & 0 & \...
- 384
5
votes
1
answer
544
views
Is the series $\sum_{n=1}^{\infty} \sin(n^4)\sin(4^n)$ convergent or divergent?
Is the series $$ \sum_{n=1}^{\infty} \sin(n^4)\sin(4^n) $$ convergent or divergent?
I tried expanding the sine functions and got no clue, and any test that I know of isn't helping me with this series. ...
- 159
5
votes
2
answers
497
views
Classification of the behaviours of the logistic map
On this this wikipedia page, it is claimed that the iterative sequence $x_{n+1}=rx_n(1-x_n)$ (the logistic map) starting at a point $[0,1]$ and where $r$ ranges in $[0,4]$ behaves differently ...
- 489
2
votes
2
answers
1k
views
Monotonic and bounded sequences throughout mathematics [closed]
When I refer to the Monotone Convergence Theorem below, I refer to the very simple claim that if a non-decreasing sequence has an upper bound then it converges. I don't refer to the claim from Measure ...
- 4,943
11
votes
2
answers
882
views
Do infinitely nested radicals have any applications?
There is a simple necessary and sufficient condition for a continued radical of the form $\sqrt{a_1 + \sqrt{a_2 + \dotsc}}$ to converge (where all terms $a_1, a_2$ etc. are nonnegative). Namely, that ...
- 4,943
26
votes
4
answers
2k
views
For $x$ irrational, is $a_{n} =\sum_{k=1}^{n}(-1)^{⌊kx⌋}$ unbounded?
For $x$ irrational, define $a_{n} :=\sum_{k=1}^{n}(-1)^{⌊kx⌋}$. Can you prove that $\left\{a_n\right\}$ is unbounded?
I feel that it is not easy to treat every irrational $x$.
I have asked in S.E. ...
- 385
13
votes
3
answers
720
views
Supremum of $ a_n = a_{n-1}^3 - a_{n-2} $
Let $a_1=0$ and let $ - \ln(2) < a_2 < \ln(2) $
Define
$$ a_n = a_{n-1}^3 - a_{n-2} $$
Then
$$ \sup_{n>2} a_n = a_2 $$
And
$$ \inf_{n>2} a_n = - a_2 $$
How to prove that ?
- 763
9
votes
0
answers
225
views
On the first sequence without collinear triple
Let $u_n$ be the sequence lexicographically first among the sequences of nonnegative integers with graphs without collinear three points (as for $a_n=n^2$ or $b_n=2^n$). It is a variation of that one.
...
57
votes
0
answers
3k
views
On the first sequence without triple in arithmetic progression
In this Numberphile video (from 3:36 to 7:41), Neil Sloane explains an amazing sequence:
It is the lexicographically first among the sequences of positive integers without triple in arithmetic ...
8
votes
0
answers
197
views
The condition on $\alpha$ that $\alpha^n$ is convergent modulo 1
We consider numbers $\alpha\in \mathbb{R}$ with $|\alpha|>1$. Is there any result about a characterization of those $\alpha$ so that $\{\alpha^n\}_{n\in \mathbb{N}}$ is convergent modulo 1?
I ...
- 661
4
votes
0
answers
93
views
Flow of zeros in the shifted exponential generating function?
Given a sequence $a_n$ (of real numbers, described more fully below), one may define the exponential generating function (on the complex plane) as $E(z)=\sum_{n=0}^\infty a_n z^n/n!$. The derivatives $...
- 384
9
votes
1
answer
236
views
Riccati-type recurrence: infinitely many sign changes?
Suppose $b_1, b_2, b_3, \dots \in \Bbb{R}$ satisfy the Riccati-type recurrence
$$b_{k+1}=\frac{1+kb_k}{k-b_k},\quad k\ge 1.$$
Is it true that such a sequence reaches infinitely many positive as ...
- 43.2k
14
votes
3
answers
586
views
Proving convergence of sum over $\mathbb{Z}^n$
In my research, I am trying to use the following construction by Benson Farb and John Franks, which proves that for all $n$, the group of $n\times n$ matrices with 1's on the diagonal, 0's above the ...
- 143
6
votes
1
answer
234
views
What about of periodic points of $\sum_{n=1}^\infty\frac{\mu(n)}{n}x^n$, $0<x<1$, where $\mu(n)$ is the Möbius function?
Let $\mu(n)$ the Möbius function, we define $F:[0,1]\to[0,1]$ as $$F(x)=\sum_{n=1}^\infty\frac{\mu(n)}{n}x^n.\tag{1}$$
For a function of this kind (I presume that this continuous function has image $[...
user75829
2
votes
0
answers
210
views
A sum with integer parts
Let $ \mathcal{A} $ be a set of reals such that $ \sum_{a \in \mathcal{A} } \frac{1}{a} = \infty $ and $ \sum_{a \in \mathcal{A} } \frac{1}{a^2} < \infty $. For instance, $ \mathcal{A} = \mathbb{N}^...
- 593
3
votes
0
answers
123
views
Irregularly Intertwined Linear Recursions: Other References?
I was wondering if anyone had run across the following notion of intertwined linear recursions. I'm looking for references, or even a standard name. (I know one source, which is the genesis of this ...
- 47.4k
0
votes
0
answers
42
views
Convergence of a positive sequence controlled by a difference inequality involving quadratic map
I have a sequence $\{x_n\}_{n\ge 0}$ with $x_0>0$, controlled by the difference inequality: $$x_{n+1}\le ax_n^2+b$$ where, $a,b>0$. Had $b$ been $0$ and $a<1$, I would find $x_n\to 0$ as $n\...
11
votes
1
answer
1k
views
Conditional convergence of $\sum_{n\geq 1} \frac{\sin(p(n))}{n}$?
The series $\sum_{n\geq 1} \frac{\sin n}{n}$ is easily seen to be conditionally convergent, e.g. by Abel summation. But how about $\sum_{n\geq 1} \frac{\sin(n^2)}{n}$? (for which Abel summation fails)...
- 462
22
votes
2
answers
1k
views
$x_1 = 2$, $x_{n + 1} = {{x_n(x_n + 1)}\over2}$, what can we say about $x_n \text{ mod }2$?
This question was asked on MathStackexchange here, but there was no answer, so I am asking it here.
Let$$x_1 = 2, \quad x_{n + 1} = {{x_n(x_n + 1)}\over2}.$$What can we say about the behavior of $x_n ...
user61522
17
votes
2
answers
1k
views
Does 53 diverge to infinity in this Collatz-like sequence?
This function has been explored a bit at MSE (in June 2016):
\begin{eqnarray}
f(n) &=& (n-1)^2 \; \textrm{if} \; (n \bmod 4) = 1\\
f(n) &=& \lfloor n/4 \rfloor \; \textrm{otherwise}
\...
- 151k
27
votes
3
answers
948
views
A point set of power series with coefficients in {-1, 1}. Connected or not?
Let $z$ be a fixed complex number with $|z|<1$ and consider the set
$$X_z := \Big\{\sum\limits_{i=1}^{\infty} a_i z^i \ \Big|\ a_i\in \{-1,1\} \forall i\Big\}.$$
What can be said about the set $M$ ...
- 373
7
votes
1
answer
4k
views
Beyond Collatz: A $5n+1$ conjecture? [closed]
Let
$$x_{n+1} = \begin{cases} x_n/2 &;\text{if } x_n \equiv 0 \pmod{2}\\ k\,x_n+1 &; \text{if } x_n\equiv 1 \pmod{2} \end{cases}$$
and $k=3$ and $x_n\in\Bbb N$. Collatz conjectured for this ...
- 254
12
votes
2
answers
1k
views
Rational numbers with dense orbits in [0,1] under iteration by f(x)=4x(1-x)
Let $f(x)=4x(1-x)$.
For which rational numbers $r\in [0,1]$ is the sequence $f^n(r)$, $n\in \mathbb N$, dense in $[0,1]$ ?
$(f^n(r)=f\circ f\circ ...\circ f(r)$ n times)
I would be happy to find a ...
- 18.7k